The point where the two lines intersect is the solution to the system.
Write the solution as a coordinate point (x, y).
Rearranging Equations
Use inverse operations to isolate the designated variable.
Solve Systems by Substitution
Steps:
Solve one of the equations for either x or y. (This step is not always necessary. Refer to the part 2 video for further direction)
Substitute the expression you have solved for into the remaining equation and solve for the lone variable.
Substitute that value into the either of original equations (or the rearranged, revised equation from step 1) and solve.
Write your answer as a coordinate point (x, y).
Part 1: When the variable is already solved for you. (Begins with step 2 from above)
Part 2: When you need to solve for a variable before substitution. (Goes through all four steps from above)
Solve Systems by Elimination
Steps:
Make sure everything is aligned and in the same column. (x's, y's, ='s, etc.)
Multiply one or both of the equations by a constant to obtain coefficients in from of one of the variables that are the same but opposite. (example: 2 and -2 or -5 and 5) (For further direction, refer to the part 2 video)
Add the equations together in columns up and down so you are combing like terms. This step will eliminate one of the variables. Solve the remaining equation.
Substitute your answer from the previous step into one of the original equations and solve for the remaining variable.
Write your answer as a coordinate point (x, y).
Part 1: When you are given the same but opposite coefficients. (skips step #2)
Part 2: When you need to multiply an equation by a number to create the same but opposite coefficients.
Special Cases
1. Same Line
When graphing the two lines will be the same.
When using algebra to solve the system (substitution or elimination), all variable will eliminate and a TRUE statement will remain (i.e. 2 = 2)
When this happens there are INFINITIELY MANY SOLUTIONS to the system
2. Parallel Lines
When graphing, the lines will never intersect. Their slopes will be the same.
When using algebra to solve the system (substitution or elimination), all variabless will eliminate and a FALSE statement will remain (i.e. 0 = 4).
When this happens there is NO SOLUTION to the system.